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In mathematics, the Laurent series of a complex function is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.
A Laurent polynomial over may be viewed as a Laurent series in which only finitely many coefficients are non-zero. The ring of Laurent polynomials R [ X , X − 1 ] {\displaystyle R\left[X,X^{-1}\right]} is an extension of the polynomial ring R [ X ] {\displaystyle R[X]} obtained by "inverting X {\displaystyle X} ".
The area of the blue region converges on the Euler–Mascheroni constant, which is the 0th Stieltjes constant.. In mathematics, the Stieltjes constants are the numbers that occur in the Laurent series expansion of the Riemann zeta function:
In polydisks, the Cauchy's integral formula holds and the power series expansion of holomorphic functions is defined, but polydisks and open unit balls are not biholomorphic mapping because the Riemann mapping theorem does not hold, and also, polydisks was possible to separation of variables, but it doesn't always hold for any domain.
The principal part at = of a function = = ()is the portion of the Laurent series consisting of terms with negative degree. [1] That is, = is the principal part of at .If the Laurent series has an inner radius of convergence of , then () has an essential singularity at if and only if the principal part is an infinite sum.
A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.
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A Laurent series is thus any series of the form ∑ n = − ∞ ∞ a n x n . {\displaystyle \sum _{n=-\infty }^{\infty }a_{n}x^{n}.} If such a series converges, then in general it does so in an annulus rather than a disc, and possibly some boundary points.